Stability and performance of stochastic economic MPC - Stochastic characterization of the closed-loop asymptotics

TL;DR

This paper provides a rigorous stochastic stability analysis of economic MPC without terminal constraints, establishing closed-loop guarantees and performance bounds using stochastic dissipativity and turnpike properties.

Contribution

It introduces a novel stochastic Lyapunov-based framework for stability and performance analysis of economic MPC without terminal conditions, extending deterministic results to stochastic settings.

Findings

Ensures $P$-practical asymptotic stability under stochastic convergence.

Derives tight bounds for average and non-averaged performance.

Shows practical implementation retains theoretical stability properties.

Abstract

Model Predictive Control (MPC) is well understood in the deterministic setting, yet rigorous stability and performance guarantees for stochastic MPC remain limited to the consideration of terminal constraints and penalties. In contrast, this work analyzes stochastic economic MPC with an expected cost criterion and establishes closed-loop guarantees without terminal conditions. Relying on stochastic dissipativity and turnpike properties, we construct closed-loop Lyapunov functions that ensure $P$-practical asymptotic stability of a particular optimal stationary process under different notions of stochastic convergence, such as in distribution or in the $p$-th mean. In addition, we derive tight near-optimal bounds for both averaged and non-averaged performance, thereby extending classical deterministic results to the stochastic domain. Finally, we show that the abstract stochastic MPC scheme requiring distributional knowledge shares the same closed-loop properties as a practically implementable algorithm based only on sampled state information, ensuring applicability of our findings. Our findings are illustrated by a numerical example.